arXiv2026
We investigate the construction and performance of summation-by-parts (SBP) operators, which offer a powerful framework for the systematic development of structure-preserving numerical discretizations of partial differential equations. Previous approaches for the construction of SBP operators have usually relied on either local methods or sparse differentiation matrices, as commonly used in finite difference schemes. However, these methods often impose implicit requirements that are not part of the formal SBP definition. We demonstrate that adherence to the SBP definition alone does not guarantee the desired accuracy, and we make additional conditions explicit that SBP operators need to satisfy in order to achieve accuracy. While these conditions are known in the SBP literature, they are usually enforced only implicitly by the respective construction procedure. Specifically, we analyze the error minimization for an augmented basis, discuss the role of sparsity, and examine the importance of nullspace consistency in the construction of SBP operators. A dispersion and dissipation analysis shows that the loss of accuracy has two sources. First, a lack of nullspace consistency produces stationary modes, which are neither transported nor damped by the scheme and whose contribution to the error converges more slowly under mesh refinement. Second, the physical mode, i.e., the discrete approximation of an exact traveling wave, can be poorly resolved, and the resulting error dominates in long-time simulations. Furthermore, we show how these design criteria can be integrated into a recently proposed optimization-based construction procedure for function space SBP (FSBP) operators on arbitrary grids. Our findings are supported by numerical experiments that illustrate the improved accuracy of the numerical solutions obtained with the proposed SBP operators.